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Let's say we have a one dimensional single particle system that is described by a SE
-[itex]\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\psi(x)+V(x)\psi(x)=E\psi(x)[/itex]
We do not know what the potential energy function V(x) is, but we know that the eigenvalues
spectrum is
[itex]E_{n}=kn^{\frac{3}{2}}[/itex]
where k is a constant with dimensions of energy and n=1,2,3,...
How to find out what is the function V(x) that results in this energy spectrum? If the exponent
were 1 instead of 3/2 then it would be a harmonic oscillator potential and if the exponent were
2 it would be the "particle in box" potential. Do all possible exponents result in a V(x) that can be
written with elementary functions?
To find V(x), we should obviously
1. Write the Hamiltonian matrix in the basis where its diagonal (we can do that)
2. Unitary transform to position representation basis
3. Subtract the kinetic energy operator
But how do we find the right unitary transformation to do this?
-[itex]\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\psi(x)+V(x)\psi(x)=E\psi(x)[/itex]
We do not know what the potential energy function V(x) is, but we know that the eigenvalues
spectrum is
[itex]E_{n}=kn^{\frac{3}{2}}[/itex]
where k is a constant with dimensions of energy and n=1,2,3,...
How to find out what is the function V(x) that results in this energy spectrum? If the exponent
were 1 instead of 3/2 then it would be a harmonic oscillator potential and if the exponent were
2 it would be the "particle in box" potential. Do all possible exponents result in a V(x) that can be
written with elementary functions?
To find V(x), we should obviously
1. Write the Hamiltonian matrix in the basis where its diagonal (we can do that)
2. Unitary transform to position representation basis
3. Subtract the kinetic energy operator
But how do we find the right unitary transformation to do this?
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